246 THE SCIENCE OF LOGIC 



such as would, in the circumstances, necessitate S being P, he would, eo ipso^ 

 have both discovered and proved S to be P, even though he had never yet 

 witnessed a single actual case of S being P. Such a mode, however, of 

 simultaneous discovery and proof, is rare in the inductive sciences, though it 

 represents, perhaps, what more usually takes place in direct, progressive, de 

 ductive lines of demonstrative inference (187, 188). 



However, it does not always happen, even in deduction, that the investi 

 gator thus simultaneously suspects, verifies, and demonstrates or explains, a 

 new truth. Let us, therefore, pursue the case in which the geometrician, for 

 instance, after finding by measurement that the square on the hypotenuse of 

 a given right-angled triangle was equal in area to the sum of the squares on 

 the sides, supposes this to be always and necessarily true of all right-angled 

 triangles, without yet knowing either that it is, or why it is, true. He sets 

 about verifying and proving his supposition not by carefully measuring a 

 number of other right-angled triangles, but rather by reflecting that if the 

 general proposition is true there must be a discoverable reason why it is true, 

 a reason which will demonstrate its truth. Hence he proceeds to seek for 

 something (Af) in the nature of S and P of the squares on the hypotenuse 

 and sides of a right-angled triangle in virtue of which S and P necessarily in 

 volve each other. This M he must discover by reflection on what he already 

 knows about triangles, squares, etc., by analysing and comparing his notions, 

 by making mental experiments, as it were, with a view to eliminating from all 

 the judgments which he finds crowding in upon his mind around &quot; 5&quot; is /*,&quot; 

 those that are unessential to the latter, until he succeeds in explaining or 

 demonstrating to himself that &quot; S is P &quot; by connecting it through the medium 

 of M with truths he has already established for certain, and so, ultimately, 

 with first principles. 



Thus, here too, as before, he verifies the new judgment &quot; S is P,&quot; or dis 

 covers that it is always true de facto, by demons trating it, by showing it to be 

 necessarily involved in already known and proved truths. In induction, on the 

 other hand, a general law is often discovered and verified long before it can be 

 explained (247). Apart from this point, however, there is a certain analogy be 

 tween the process just outlined and the experimental testing and verification of 

 an hypothesis in induction. Both are illustrations of what the Schoolmen called 

 &quot; inventio medii&quot; the process of finding a &quot; middle term &quot; of proof (167, 197). 

 In induction, having observed 6&quot; to be de facto connected with P in some case 

 or cases, our task is to find out whether the connexion is a necessary, causal 

 connexion, or only an accidental, contingent one ; and we prove that it is causal 

 by showing (if we can) that something essential to S, namely M, is necessarily 

 or causally connected with P. But how do we satisfy ourselves here that the 

 M which connects S with P is itself necessarily connected with P ? In other 

 words, how do we know that our premisses (especially our major premiss, &quot; M 

 is P &quot;) are necessarily and universally true ? Not as in the deductive sciences, 

 by showing our premisses to be either intrinsically self-evident principles or else 

 derived by demonstration from such principles, but, as we saw in dealing with 

 the verification of hypotheses (229-33) by convincing ourselves, through obser 

 vation and experiment, that nothing else but the truth of those premisses is 

 compatible with the conclusion which we know to be true as a fact. In other 

 words, we have to convince ourselves, by inductive investigation, that those pre- 



